Empirical validation of structural metrics for predicting understandability of conceptual schemas...

ORIGINAL ARTICLE

Empirical validation of structural metrics for predictingunderstandability of conceptual schemas for data warehouse

Manoj Kumar • Anjana Gosain • Yogesh Singh

Received: 28 August 2012 / Revised: 24 March 2013

� The Society for Reliability Engineering, Quality and Operations Management (SREQOM), India and The Division of Operation and

Maintenance, Lulea University of Technology, Sweden 2013

Abstract Data warehouse (DW) quality depends on its

data models (conceptual, logical and physical model).

Multidimensional (MD) modeling has been widely recog-

nized as the backbone of data modeling for DW. Recently,

some of the authors have proposed a set of structural

metrics to assess quality of MD conceptual models. They

have found the significant relationship between metrics and

understandability of DW conceptual schemas using various

correlation analysis techniques such as Spearman’s, Pear-

son etc. However, advanced statistical and machine learn-

ing methods have not been used to predict effect of each

metric on understandability. In this paper, our focus is on

predicting the effect of structural metrics on understand-

ability of conceptual schemas using (i) statistical method

(logistic regression analysis) that include univariate and

multivariate analysis, (ii) machine learning methods

(Decision Trees, Naive Bayesian Classifier) and (iii)

compare the performance of these statistical and machine

learning methods. The results obtained show that some of

the metrics individually have a significant effect on the

understandability of MD conceptual schema. Further, few

of the metrics have a significant combined effect on

understandability of conceptual schema. The results also

show that the performance of Naive Bayesian Classifier

prediction method is better than logistic regression analysis

and Decision Trees methods.

Keywords Data warehouse quality � Multidimensional

conceptual model � Metrics � Logistic regression analysis �Naive Bayes Classifier � Decision Trees

1 Introduction

Data warehouses (DW) are large repositories created to

hold data extracted from various sources and maintained by

different operating units together with historical and sum-

mary transformations. Data warehouses are the core of

most of the current decision support systems and provide

historical information for decision making process. Due to

increased complexity of DW (Inmon 2003), continuous

attention must be paid for evaluation of quality throughout

its design and development (Bouzeghoub and Kedad 2002;

Serrano et al. 2008). A lack of quality in DW may have

disastrous impacts from both technical and organizational

point of view such as loss of clients and financial losses or

discontent among employees (English 1996). Therefore, it

becomes important for an organization to guarantee the

quality of information maintained in its DW.

According to Serrano et al. (2007), information quality

of data warehouse is determined by (i) the quality of the

DW itself and (ii) the quality of data presentation as shown

in Fig. 1. Data warehouse quality can be influenced by data

base management systems quality, data quality and data

model quality. Thus, one of the main issues that influence

M. Kumar (&)

Department of Computer Science & Engineering, Ambedkar

Institute of Advanced Communication Technologies &

Research, Delhi, India

e-mail: [email protected]

A. Gosain

University School of Information Technology, Guru Gobind

Singh Indraprastha University, Delhi, India


Y. Singh

The Maharaja Sayajirao University of Baroda,

Vadodara, Gujarat, India


123

Int J Syst Assur Eng Manag

DOI 10.1007/s13198-013-0159-4

DW quality depends on the data models (conceptual, log-

ical and physical, see Fig. 1). Here, in this paper our focus

will be on assessing the understandability of data ware-

house conceptual model, as we can refer Fig. 2, under-

standability affects quality of DW models.

Multidimensional (MD) modeling has been widely recog-

nized as the backbone of data modeling for DW. Some

approaches (Kimball and Ross 2002; Inmon 2003; Vas-

siliadis 2000) have been lately proposed for logical and

physical modeling of DW. They also proposed various

guidelines for creating good multidimensional models.

However, these guidelines or subjective assessment

criteria were not sufficient to guarantee the quality of DW

data model. Further, various approaches (Abello et al.

2002; Golfarelli et al. 1998; Golfarelli and Rizzi 1998;

Sapia 1999; Tryfona et al. 1999) have also been proposed

for conceptual modeling of data warehouse. Unfortunately,

none of these approaches considers neither subjective nor

objective quality of conceptual schema.

Recently, a set of structural metrics have been defined

for assessing quality of DW conceptual model (Serrano

2004; Serrano et al. 2007). The purpose of these metrics is

to facilitate designers while choosing among alterative

schemas that are semantically equivalent. Obtaining a valid

set of metrics is not only a matter of definition but also two

kinds of metrics validation are required, theoretical and

empirical. Theoretical validation is used to verify analyti-

cally that the metrics are proper numerical characteriza-

tions of the measured attribute (conforming to a set of

formal properties). Such validation has been conducted on

DW metrics (Serrano 2004). They have found the rela-

tionship between metrics and understandability of DW

conceptual schemas using various correlation analysis

techniques such as Spearman’s, Pearson etc. However,

advanced statistical and machine learning methods have

not been used to predict the effect of each metric on

understandability. In contrast, these methods have been

applied to predict software fault proneness using object

oriented metrics (Singh et al. 2010; Malhotra et al. 2010).

In this paper, our focus is on predicting the effect of

structural metrics on understandability of conceptual

schemas using:

(i) Statistical method (logistic regression analysis) that

include univariate and multivariate analysis.

(ii) Machine learning methods (Decision Trees, Naive

Bayesian Classifier).

(iii) Compare the performance of these statistical and

machine learning methods.

Organization of this paper is as follows: in Sect. 2, we

discuss multidimensional modeling and quality metrics for

data warehouse. In Sect. 3, metrics for DW conceptual

Fig. 1 Data warehouse

information quality (Serrano

et al. 2007)

Fig. 2 Relationship between

structural properties, cognitive

complexity, understanding and

external quality attributes

(Serrano et al. 2007)


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model are discussed. The Sect. 4 discusses the experi-

mental design that include, the independent and dependent

variables, hypothesis to be tested, and the data collected for

our study. In Sect. 5, we discuss statistical (logistic

regression analysis) and machine learning methods (Naive

Bayesian Classifier, Decision Trees) as research method-

ology for our study. In Sect. 6, we discuss the analysis

results using both statistical and machine learning methods

for prediction of understandability of conceptual schema.

In Sect. 7, we compare the performance of statistical and

machine learning methods for predicting understandability

of conceptual schema. The Sect. 8 presents validation of

proposed hypothesis. In Sect. 9, we present threats to

validity of our obtained results followed by conclusion and

future work in Sect. 10.

2 Related work

In this section, we organize the related work regarding two

relevant research topics: (i) multidimensional modeling for

DW, and (ii) the quality issues in data warehouse metrics.

2.1 Multidimensional (MD) modeling

Various multidimensional data models have been proposed.

Some of the models were proposed at the logical level such as

the well known star schema proposed by Kimball and Ross

(2002). Others may be considered as formal models as they

provide a formalism to consider main MD properties. The

detailed review of the most relevant logical and formal

models can be found in Blaschka et al. (1998) and Abello

et al. (2001). Various, pure conceptual MD models have also

been witnessed in the DW literature (Abello et al. 2002;

Golfarelli et al. 1998; Golfarelli and Rizzi 1998; Sapia et al.

1998; Sapia 1999; Tryfona et al. 1999; Trujillo et al. 2001).

These models provide a high level of abstraction for the main

MD modeling properties at the conceptual level and are

totally independent from implementation issues. One out-

standing feature provided by these models is that they pro-

vide a set of graphical notations (such as the classical and

well known EER model) that facilitates their use and reading.

These are as follows: the Dimensional-Fact (DF) Model by

Golfarelli et al. (1998), the Multidimensional/ER (M/ER)

Model by Sapia et al. (1998), the starER Model by Tryfona

et al. (1999), the Model proposed by Husemann et al. (2000)

and the Yet Another Multidimensional Model (YAM2) by

Abello et al. (2002). Unfortunately, none of these has been

accepted as standard for the conceptual modeling of DW.

Recently, another approach (Lujan-Mora et al. 2002;

Trujillo et al. 2001) has been proposed as an object-

oriented (OO) conceptual MD modeling approach. This

proposal was a profile of the Unified Modeling Language

(UML) (OMG 2005), which used the standard extension

mechanisms (stereo-types, tagged values and constraints)

provided by the UML. In (Lujan-Mora et al. 2002), the

conceptual model has been represented by means of a class

diagram in which the information is organized in facts and

dimensions, as shown in Fig. 3. Some of the characteristics

that can be represented in this model (Lujan-Mora et al.

2002) are the relationships many-to-many between the

facts and one specific dimension, the degenerated dimen-

sions, the multiple classification and alternative path hier-

archies, and the non-strict and complete hierarchies

(Cherfil and Prat 2003). Here, facts may be represented by

fact classes and dimension by dimension classes. In Fig. 3,

we are interested in analyzing the car sales (Fact Car Sales)

of a big showroom. This fact ‘car sales’ contains the spe-

cific measures to be analyzed, i.e., quantity and price. On

the other hand, the main dimensions along with we would

like to analyze these measures are the Time, the specific

model sold, the customer and the sales representative.

Finally, Base classes Week, Quarter and Year; and City

and Country represent the classification hierarchies of the

Time and Customer dimensions, respectively.

However, none of these approaches for MD modeling

considers the quality of conceptual schemas as an important

issue of their models and they do not neither subjective nor

objective (metrics) indicators (Serrano et al. 2007). In the

next section, we discuss the quality aspect of data warehouse.

2.2 Quality issues in data warehouse metrics

It is very important to consider quality issues at various

levels of data models: conceptual, logical and physical

models (Serrano et al. 2007). Different approaches have

been researched for optimizing physical data models (Ha-

rinarayan et al. 1996; Labio et al. 1997), whereas quality of

the data warehouse logical models (Serrano et al. 2002;

Serrano 2004) have been assured by proposing several

metrics to evaluate the quality of star schemas at logical

level. They validated their suggested metrics theoretically

as well as empirically.

Later, the quality of conceptual models for data ware-

houses has been discussed in (Jarke et al. 2002). Nevertheless,

this model only considers quality as intuitive notion. In this

manner, it is not easy to guarantee the quality of DW con-

ceptual models, a problem which has been addressed by

Jeusfeld et al. (1998) in the context of DWQ project. This

direction of research addresses the definition of metrics that

allows us to replace the intuitive notions of quality regarding

the conceptual model of data warehouses with formal and

quantitative measures. The efforts in this direction, includes

normal forms for DW design as initially proposed in (Lehner


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et al. 1998) and generalized in (Lechtenborger and Vossen

2003). These normal forms represented a first step towards

objective quality metrics for conceptual schemata. Further,

Cherfil and Prat (2003) have proposed metrics for measuring

multidimensional schemas analyzability and simplicity.

However, the metrics proposed in these approaches have not

been empirically validated and consequently, have not proven

their practical applications (Fenton and Pfleeger 1997).

Recently, Serrano (2004) and Serrano et al. (2007) proposed a

set of metrics for assessing the understandability of data

warehouse conceptual schemas and also validated the metrics

theoretically and empirically through a family of experi-

ments. However, in their empirical validation, they have not

used advanced statistical and machine learning methods to

predict understandability of DW conceptual schemas using

structural metrics. In this paper, we will use these prediction

Fig. 3 UML based DW

conceptual model depicting

CAR sales

Table 1 Metrics definition

Metrics Definition

NDC(S) Number of dimension classes of the star S (equal to the number of aggregation relationships)

NBC(S) Number of base classes of Star S

NC(S) Total number of classes of the star S NC(S) = NDC(S) ? NBC(S) ? 1

RBC(S) Ratio of base classes. Number of base classes per dimension class of the star S

NAFC(S) Number of FA attributes of the fact class of the star S

NADC(S) Number of D and DA attributes of the dimension classes of the star S

NABC(S) Number of D and DA attributes of the base classes of the star S

NA(S) Total number of FA, D and DA attributes of the star S NA(S) = NAFC(S) ? NADC(S) ? NABC(S)

NH(S) Number of hierarchy relationships of the star S

DHP(S) Maximum depth of the hierarchy relationships of the star S

RSA(S) Ratio of attributes of the star S. Number of attributes FA divided by the number of D and DA attributes


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methods (Statistical and Machine Learning) to predict

understandability of conceptual schemas using structural

metrics (Serrano 2004; Serrano et al. 2007), which are defined

in the next section.

3 Metrics for data warehouse conceptual model

Following are the metrics definition (Serrano 2004; Serrano

et al. 2007) for measuring the understandability of data

warehouse conceptual models with fact classes, dimen-

sional classes and associated base classes and their attri-

butes (Table 1).

4 Experimental design

In this section, we present the variables in the study along

with hypothesis to be tested and finally the data collected

for this study.

4.1 Variables in the study

4.1.1 Independent variables

The independent variables are the variables for which the

effects should be evaluated for predicting the understand-

ability of DW conceptual schema. In our study,

independent variable corresponds to the structural com-

plexity, which is measured through the metrics being dis-

cussed in Sect. 3. Here, we have considered 20 real world

examples of an object oriented data warehouse conceptual

schemas using UML and calculated the values of the

metrics, as shown in Table 2.

4.1.2 Dependent variables

Understandability is the dependent variable, which we

want to predict using structural metrics (independent

variables). The understandability of the tests was measured

as the time each subject took to perform the tasks of each

experiment test. The experimental tasks consisted of

understanding the schemas and answering the corre-

sponding questions. Here, low value of understanding time

for the schema means, it is understandable, whereas, high

value of understanding time for the schema means, it is

non-understandable.

4.2 Hypothesis formulation

In this section, research hypothesis are presented.

Null hypothesis, H0 There is no significant effect of

individual metric on schema understandability.

Alternate hypothesis, H1 There is a significant effect of

NDC metric on schema understandability.

Table 2 Collected metrics

Metrics schema NDC NBC NC RBC NAFC NADC NABC NA NH DHP RSA

S01 4 4 9 1 2 18 4 24 3 3 0.09

S02 3 7 11 2.34 1 13 7 21 3 3 0.05

S03 4 4 9 1 3 17 4 24 2 3 0.14

S04 4 0 5 0 2 23 0 25 0 0 0.08

S05 4 4 9 1 2 20 4 26 2 3 0.08

S06 3 7 11 2.34 2 13 7 22 3 3 0.1

S07 4 5 10 1.25 2 12 5 19 2 3 0.11

S08 3 7 11 2.34 2 15 7 24 2 4 0.09

S09 5 8 14 1.6 2 22 8 32 3 3 0.06

S10 5 4 10 0.8 2 13 4 19 2 2 0.11

S11 5 4 10 0.8 2 21 4 27 2 3 0.08

S12 5 4 10 0.8 2 22 4 28 2 3 0.07

S13 3 4 8 1.37 2 15 4 21 2 3 0.10

S14 3 4 8 1.37 2 16 4 22 2 2 0.1

S15 4 2 7 0.5 2 14 2 18 1 2 0.12

S16 3 0 4 0 4 15 0 19 0 0 0.26

S17 5 7 12 1.4 2 18 7 27 3 4 0.08

S18 4 2 6 0.5 3 12 2 17 1 2 0.21

S19 4 6 10 1.5 2 15 6 23 3 2 0.09

S20 5 3 8 0.6 2 27 3 32 1 3 0.06


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NBC metric on schema understandability.


NC metric on schema understandability.


RBCC metric on schema understandability.


NAFC metric on schema understandability.


NADC metric on schema understandability.


NABC metric on schema understandability.


NA metric on schema understandability.


NH metric on schema understandability.


DHP metric on schema understandability.


RSA metric on schema understandability.

Alternate hypothesis, H12 There is a significant com-

bined effect of some metrics on schema understandability.

4.3 Empirical data collection

There were 18 subjects, who were doing M. Tech. at the

Institute, where one of the author is employed. These 18

subjects have participated in the experiment. We gave a set

of questions for each schema to all the subjects. The sub-

jects had to analyze each schema and answer related

questions of that particular schema. In this experiment we

have taken fix number of questions in each schema, for

instance a set of four questions for schema car sales

(Fig. 3) are given as follows.

(1) Which classes do we need to use for knowing the

model of the car?

(2) Which classes do we need to know that which sales

representative has helped in the maximum sales?

(3) Which classes do we need to answer the total sales

price in a year?

(4) If we want to increase our car sales then in which

class do we add our promotion advertisement?

The starting time and finish time in which the subjects

answers these questions for each schema is noted down in

seconds and we get the understanding time i.e. time taken

to solve the questions. The collected understanding time of

18 subjects for 20 conceptual schemas for DW taken from

various books of Data warehousing are shown in the

Table 3.

5 Research methodology

In this section, we discuss both statistical (logistic regres-

sion analysis) and machine learning methods (Decision

Trees, Naive Bayesian Classifier) for predication of schema

understandability.

5.1 Statistical method

The logistic regression (LR) analysis is used to predict the

dependent variable (understandability) from a set of inde-

pendent variables (structural metrics) in order to determine

the percentage of variance in the dependent variable

explained by the independent variable (Basili et al. 1996;

Hosmer and Lemeshow 1989). It is used when the outcome

variable is binary or dichotomous. We have used both

univariate and multivariate regression. Univariate logistic

regression finds the relationship between the dependent

variable and each independent variable. It finds whether

there is any significant association between them. Multi-

variate logistic regression is done to construct a prediction

model for the understandability of multidimensional con-

ceptual schema. It analyzes which metrics are useful when

they are used in combination. Logistic regression results in

a subset of metrics that have significant parameters. To find

the optimal set of independent variables (metrics), there are

two stepwise selection methods, which are forward selec-

tion and backward elimination (Hosmer and Lemeshow

1989). Forward selection examines the variables that are

selected one at a time for entry at each step. The backward

elimination method includes all the independent variables

in the model and the variables are deleted one at a time

from the model until the stopping criteria is fulfilled. We

have used the forward stepwise selection method. The

general multivariate logistic regression formula (Basili

et al. 1996; Hosmer and Lemeshow 1989; Aggarwal et al.

2009), which we used is as follows:

Prob X1; X2; . . .; Xnð Þ ¼ egðxÞ

1þ egðxÞ

where g(x) = B0 ? B1*X1 ? B2* X2 ? … ? Bn*Xn,

‘prob’ is the probability of a schema being understandable

and Xi, (1 B i B n) are independent variables.

The following statistics are reported for each metric

from the above formula:

• Odds ratio: The odds ratio is calculated using Bi’s. The

formula for the odds ratio (R) = exp (Bi). This is

calculated for each independent variable. The odds


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Table 3 Collected understanding time of 18 subjects for 20 schemas in seconds

Schemas Understanding time of subjects (s)

Subj1 Subj2 Subj3 Subj4 Subj5 Subj6 Subj7 Subj8 Subj9

S01 60 70 72 58 64 84 74 73 65

S02 45 40 56 49 35 41 51 32 29

S03 52 45 51 47 40 36 54 41 62

S04 35 40 41 36 32 31 28 25 30

S05 50 53 51 46 56 65 74 29 35

S06 48 39 26 28 37 35 40 27 36

S07 32 31 38 26 28 40 34 50 51

S08 63 68 75 74 80 59 65 71 78

S09 81 86 100 89 79 67 71 86 74

S10 56 68 78 45 61 50 53 51 48

S11 46 53 52 48 70 68 61 55 42

S12 45 53 55 61 60 59 57 45 39

S13 40 35 39 25 29 34 31 41 28

S14 22 27 36 31 29 34 32 33 24

S15 32 37 39 41 45 29 24 26 33

S16 38 40 48 36 22 24 29 35 34

S17 86 74 96 102 68 67 88 77 84

S18 50 45 46 38 29 27 36 37 47

S19 56 58 60 84 74 63 41 45 50

S20 68 74 77 84 63 59 69 78 81

Schemas Understanding time of subjects (s)

Subj10 Subj11 Subj12 Subj13 Subj14 Subj15 Subj16 Subj17 Subj18

S01 50 54 70 60 71 54 57 67 55

S02 47 48 44 54 48 38 50 45 51

S03 32 47 40 50 60 38 42 50 43

S04 26 28 32 35 40 33 42 38 31

S05 44 46 34 70 53 50 42 37 54

S06 22 26 32 38 47 30 35 32 28

S07 41 40 28 32 37 52 30 38 50

S08 69 60 55 72 76 63 58 75 56

S09 89 85 65 78 90 96 87 79 92

S10 89 58 66 76 60 52 51 49 47

S11 40 42 46 63 65 71 52 68 45

S12 41 43 52 59 62 47 40 50 55

S13 27 29 30 32 35 40 39 26 32

S14 22 25 30 34 28 26 25 32 35

S15 38 33 36 38 40 27 28 26 40

S16 27 29 32 36 41 24 26 34 45

S17 70 67 79 95 80 77 68 73 83

S18 45 44 48 36 37 30 44 48 43

S19 61 59 49 40 62 72 82 67 51

S20 88 85 83 67 62 66 81 75 72


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ratio is the probability of the event divided by the

probability of a non-event. The event in our study is the

probability of having a schema understandable and the

non-event is the probability of not having a under-

standable schema.

• Maximum likelihood estimation (MLE) and coefficients

(Bi’s): MLE is the likelihood function that measures the

probability of observing a set of dependent variables.

MLE finds the coefficient in such a way that the log of

the likelihood function is as large as possible. The more

the value of the coefficient the more the impact of the

independent variables on predicted understandability of

the schema is.

• The statistical significance: Statistical significance mea-

sures the significance level of the coefficient. The larger the

value of the statistical significance (Sig.) the lower the

estimated impact of the independent variables (metrics).

• The R2 statistic: It is the proportion of the variance in

the dependent variable (understandability) that is

explained by the variance of independent variables

(metrics). The higher is the value of R2, the higher is

the effect of the independent variables and more is the

accuracy of the model.

5.2 Machine learning methods

The machine learning techniques deal with the issues of

how to build and design computer programs that improve

their performance for some specific task based on past

observations. In this section we discuss, well known

machine learning methods (Decision Trees, Naive Bayes-

ian Classifier) for prediction of understandability of con-

ceptual schema.

5.2.1 Decision Trees method

A decision tree (DT) is a predictive machine learning

model that decides the target value (dependent variable) of

a new sample based on various attribute values of the

available data. The DT includes several algorithms such as

Quinlan’s ID3, C4.5, C5, J48 and CART (Ross 1993;

Kohavi 1995; Witten and Frank 2011; Han and Kamber

2007). We discuss the J48 algorithm, which is a modified

version of an earlier algorithm C4.5 developed by Ross

(1993). J48 algorithm builds Decision Trees from a set of

training data using the concept of information entropy. The

training data is a set S = s1, s2, s3,—of already classified

samples. Each sample si = x1, x2, x3,… is a vector where

x1, x2, x3,… represent attributes or features of the sample.

The training data is augmented with a vector C = c1, c2,

c3,—where c1, c2, c3,—represent the class to which each

sample belongs. At each node of the tree, J48 chooses one

attribute of the data that most effectively splits its set of

samples into subsets enriched in one class or the other. Its

criterion is the normalized information gain that results

from choosing an attribute for splitting the data. The

attribute with the highest information gain is chosen to

make the decision. The J48 recursively classifies the tree

until each leaf is pruned, i.e. the data has been categorized

as close to perfect as possible (Witten and Frank 2011).

Moreover, among all possible values of information gain, if

there is any other value for which there is no ambiguity

then we terminate that branch and assign it to the target

value that we have obtained so far. Alternatively, we look

for another attribute providing highest information gain.

Therefore we continue this process until we either get a

clear decision of what combination of attributes provide us

a particular target value, or we run out of attributes. In such

event when we run out of attributes, or if we cannot get an

unambiguous result from the available training set, then we

assign this branch a target value that the majority of items

under this branch possess. Now by checking all these

respective attributes and their values observed in DT, we

can predict the target value of new instances (Ross 1993).

5.2.2 Naive Bayesian classification method

Naive Bayesian Classifier can predict class membership

probabilities, such as the probability that a given tuple

belongs to a particular class (Kohavi 1995; Han and

Kamber 2007). It assumes that the effect of an attribute

value on a given class is independent of the value of other

attributes. This assumption is called conditional indepen-

dence, which is explained as follows:

P HjXð Þ ¼ P XjHð ÞP Hð Þ=P Xð Þ

P(H|X) reflects the probability that the conceptual

schema X will be understandable given that we know the

value of metrics of schema X.

P(H) is the prior probability of H. For instance, this is

the probability that any given conceptual schema will be

understandable, regardless of its metrics values, or any

other information for that matter.

Similarly, P(X|H) is the posterior probability of X

conditioned on H. That it is the probability that the value of

metrics NDC and NA of conceptual schema X are 4 and 24,

respectively given that the conceptual schema X is

understandable.

P(X) is the prior probability of conceptual schema X. It

is the probability that a particular conceptual schema from

a set of conceptual schemas has value 4 and 24 of its

metrics NDC and NA, respectively.

• Let D be a training set of tuples and their associated

class labels. As usual each tipple is represented by


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n-dimensional attribute vector, X = (x1, x2,…xn),

depicting n measurements made on the tuple from n

attributes, respectively, A1 A2, A3,…An.

• Suppose that there are m classes c1, c2,…cm Given a

tuple X, the classifier will predict that X belongs to the

class having the highest posterior probability, condi-

tioned on X. That is, the Naı̈ve Bayesian Classifier

predicts that tuple X belongs to the class ci, if and only

if

P cijxð Þ[ P cjjx� �

for i � j � m; j 6¼ i:

Thus we maximize P(ci|x). The class ci for which

P(ci|x) is maximized is called the maximum posterior

hypothesis

P cijxð Þ ¼ P xjcið ÞP cið Þ=P xð Þ

• As P(x) is constant for all classes, only P(x|ci) P(ci)

need to be maximized. If the class prior probabilities

are not known than it is commonly assumed that the

classes are equally likely, that is P(c1) = P(c2) =

P(cm), and we would therefore maximize P(x|ci).

Otherwise, we maximize P(x|ci) P(ci).

The structure of Naive Bayesian Classifier represents

knowledge in the form of a tree which is constructed

recursively. The leaf nodes are Naive Bayes categorizers

for predicting a single class. We employed tenfold cross

validation on our collected data. Naive Bayesian algorithm

tries to approximate whether the generalization accuracy of

Naive Bayes at each leaf is higher than a single Naive

Bayes Classifier at the current node (Kohavi 1995).

6 Analysis results

In this section, we described the analysis performed to find

the effect of each metric on understandability of conceptual

schema. We applied univariate and multivariate logistic

regression analysis as well as two machine learning

methods (Decision Trees, Naive Bayesian Classifier).

Following measures are used to evaluate the performance

of each predicted understandability model:

6.1 Sensitivity

It measures the correctness of the predicted model. It is

defined as the percentage of schemas correctly predicted to

be understandable.

Sensitivity = [(Number of schemas correctly predicted

as understandable)/(total number of actual understandable

schemas)]*100

6.2 Specificity

It also measures the correctness of the predicted model. It

is defined as the percentage of schemas predicted that will

not be understandable.

Specificity = [(Number of schemas correctly predicted

as non-understandable)/(total number of actual non

understandable schemas)]*100

6.3 Accuracy

It is defined as the ratio of number of schemas (including

understandable and non-understandable) that are predicted

correctly to the total number of schemas.

Accuracy = {[(Number of schemas correctly predicted

as understandable) ? (number of schemas correctly pre-

dicted as non-understandable)]/total schemas}*100

6.4 Receiver operating characteristic (ROC) analysis

The performance of the outputs of the predicted models

may be evaluated using ROC analysis. It is an effective

method of evaluating the performance of the model pre-

dicted. The ROC curve is defined as a plot of sensitivity on

the y-coordinate versus its 1-specificity on the x-coordinate

(El Emam et al. 1999). The area under ROC curve (AUC)

is a combined measure for accuracy of prediction model.

The validation method used in our study is k-cross vali-

dation (the value of k is taken as 10) in which the dataset is

divided into approximately equal k partitions (Stone 1974).

One partition at a time is used for testing the model and the

remaining k - 1 partitions are used for training the model.

This is repeated for all the k partitions.

6.5 Descriptive statistics

We have shown the maximum, minimum, standard devia-

tion and average of understandability time taken by the

subjects to answer the given set of questions for each

schema of the experiments, as shown in the Table 4.

Following observation are made from the Table 4:

• For schema (S17), the subject taken highest maximum

(102 s), highest minimum (67 s), highest standard

deviation (11.70 s) and highest average (81.05 s)

understanding time among all schemas.

• For schema (S14), the subject taken lowest maximum

(40 s), lowest minimum (22 s), lowest standard devi-

ation (5.34 s) and lowest average (30.61 s) understand-

ing time among all schemas.

• The average of all the average understanding time of

each schema is 50.56 s. We believe that, if average

understanding time of the schema is less than or equal


123

to 50.56 s, than schema is understandable otherwise

schema is non-understandable. In this way, understand-

ability attribute (dependent variable) will be trans-

formed from continuous to categorical variable.

6.6 Univariate LR analysis results

In univariate LR analysis, effect of each metrics on

the understandability is predicted and shown in the

Table 5:

NDC, NC and NA metrics are found to be significant

since their significance level is less than 0.054. The impact

of the metrics NDC, NC and NA on predicting under-

standability is negative, since value of coefficients (B) are

negative. That means, more value of these metrics for a

conceptual schema may lead to less understandability (i.e.

understanding time would be more). The NDC metric has

the highest R2 value, which shows that it is the best pre-

dictor for understandability of conceptual schema. The NA

metric has the second highest R2 value. The values of R2

statistic are more important as compared to the value of

Sig., as they show the strength of correlation.

6.7 Multivariate LR analysis results

In this section, we summarize the result obtained using LR

analysis method. The multivariate analysis is used to deter-

mine the combined effect of structural metrics on under-

standability of conceptual schema. We used the forward

stepwise selection method. The result is shown in Table 6,

Table 4 Descriptive statistics

of data collected from the

experiment

Schema Understanding time (s) Understandability

Maximum Minimum SD Average

S01 84 48 9.90 64.94 Non-understandable

S02 56 28 8.86 41.44 Understandable



















Table 5 Logistic model correlation coefficients (B), odd ratio and R2

statistics

Metric B SE Sig. Odd ratio R2

NDC -2.448 1.0628 0.0213 0.0865 0.3903

NBC -0.4039 0.2539 0.1117 0.6677 0.1434

NC -0.6981 0.3605 0.0531 0.4975 0.2845

RBC -0.3225 0.67 0.6349 0.7244 0.0113

NAFC 1.0138 1.039 0.3260 2.75 0.0603

NADC -0.2434 0.1375 0.0766 0.784 0.1789

NABC -0.4039 0.25 0.1117 0.6677 0.1441

NA -0.4141 0.1911 0.0302 0.6609 0.3366

NH -0.9847 0.63 0.1181 0.3736 0.1439

DHP -1.154 0.75 0.1200 0.315 0.174

RSA 40.9831 25.91 0.1138 629142582 0.2345

Table 6 The LR analysis method statistics

Metric B SE Sig. Odd ratio R2

NDC -3.4951 2.0424 0.087 0.0303

NC -1.5134 1.1770 0.1985 0.2202

Constant 28.2333 18.3118 0.1231 182619600724.68 0.551


123

which shows that two metrics NDC and NC are included in the

multivariate model. The sensitivity, specificity, precision etc.

for each metrics and model predicted are shown in Table 7.

NDC metric has the highest sensitivity (100 %), speci-

ficity (66.60 %) and accuracy (85 %) for prediction of

understandability of conceptual schema. The NA metric has

the second highest sensitivity (81.8 %), specificity (55.50 %)

and accuracy (70 %) for prediction of understandability of

conceptual schema. However, specificities and accuracy of

other metrics are lesser than NDC and NA metrics. The

sensitivity, specificity and the accuracy of the understand-

ability prediction model are 90.9, 77.7 and 85.7 %, respec-

tively. The value R2 of the predicted model is 0.551

6.8 Decision Trees (DT) analysis results

The NDC metric has highest sensitivity (100 %), speci-

ficity (66.66 %), and highest accuracy (85 %) for

prediction of understandability of DW conceptual schema.

The NA metrics has also the highest sensitivity (100 %),

second-highest specificity (55.55 %), and second highest

accuracy (80 %) for prediction. However, all other metrics

have lesser accuracy than NDC and NA metrics. The pre-

diction model has the highest sensitivity (100 %), speci-

ficity (88.88 %) and accuracy (95 %) (Table 8).

6.9 Naive Bayesian Classifier (BC) analysis results

The NDC metric has highest sensitivity (100 %), second-

highest specificity (66.66 %), and highest accuracy (85 %)

for prediction of understandability of DW conceptual

schema. The RSA metric has second-highest sensitivity

(81.81 %), highest specificity (88.88 %), and highest

accuracy (85 %) for prediction. Both NA and NC metrics

has third-highest sensitivity (72.72 %), second-highest

specificity (77.77 %), and second-highest accuracy (75 %)

for prediction. However, all other metrics have lesser

accuracy than NDC, NC, NA and RSA metrics. The pre-

diction model has the sensitivity of 81.81 %, specificity of

100 %, and accuracy of 90 % (Table 9).

7 Comparison of prediction methods

We have applied statistical method (LR analysis) and

machine learning methods (Decision Trees, Naive Bayes-

ian Classifier) to predict understandability of DW con-

ceptual schema using structural metrics. The method is

called better for prediction, if it reports high sensitivity and

high specificity during prediction of understandability of

DW conceptual schema. We have evaluated the perfor-

mance of each method through tenfolds cross validation.

The performance of LR analysis, Decision Trees and Naive

Table 7 Sensitivity, specificity, accuracy and AUC

Metric Sensitivity Specificity Accuracy AUC

NDC 1 0.666 0.85 0.853

NBC 0.615 0.571 0.6 0.686

NC 0.818 0.333 0.6 0.789

RBC 0.545 0.444 0.5 0.585

NAFC 0.909 0 0.5 0.591

NADC 0.727 0.666 0.7 0.747

NABC 0.727 0.444 0.6 0.686

NA 0.818 0.555 0.7 0.838

NH 0.818 0.444 0.65 0.697

DHP 1 0.222 0.3 0.697

RSA 0.727 0.555 0.65 0.773

Model 0.909 0.777 0.857 0.934

Table 8 DT analysis: sensitivity, specificity, precision, accuracy and

AUC


NDC 1 0.666667 0.85 0.833

NBC 0.363636364 1 0.65 0.682

NC 0.363636364 1 0.65 0.682

RBC 1 0.444444 0.75 0.823

NAFC 1 0 0.55 0.5

NADC 0.818181818 0.666667 0.75 0.742

NABC 0.363636364 1 0.65 0.682

NA 1 0.555556 0.8 0.778

NH 1 0 0.55 0.5

DHP 1 0.222222 0.65 0.611

RSA 0.727272727 0.888889 0.8 0.808

Model 1 0.888889 0.95 0.985

Table 9 Naive BC analysis: sensitivity, specificity, accuracy and

AUC


NDC 1 0.666667 0.85 0.854

NBC 0.363636364 0.888889 0.6 0.687

NC 0.727272727 0.777778 0.75 0.788

RBC 0.545454545 0.444444 0.5 0.626

NAFC 0.363636364 1 0.65 0.682

NADC 0.818181818 0.444444 0.65 0.747

NABC 0.363636364 0.888889 0.6 0.687

NA 0.727272727 0.777778 0.75 0.848

NH 0.363636364 0.888889 0.6 0.697

DHP 0.181818182 1 0.55 0.682

RSA 0.818181818 0.888889 0.85 0.848

Model 0.818181818 1 0.9 0.96


123

Bayesian Classifier (BC) are shown in ROC curves of

Figs. 4, 5, and 6, respectively. The area under ROC curve

(AUC) is a combined measure for accuracy of prediction

model. The method that reports AUC value closer to 1 is

said to have good method for more accurate prediction.

The results of each prediction method applied are also

shown in Table 10: The decision tree method shows

highest sensitivity of 81.81 %, whereas Naive Bayesian

Classifier (BC) method shows highest specificity of

77.77 %. Further, Naive BC method reports prediction

accuracy of 75 % accuracy, which is same as reported by

the DT method. The Naive BC method reports highest

AUC value of 0.889 amongst DT and LR analysis method.

Therefore, on the basis of AUC value, we can conclude that

the performance of Naive BC method is better than LR

analysis and decision tree method for predicting the

understandability of DW conceptual schemas.

8 Validation of hypothesis and discussion

In this section, we validate our hypothesis formulated in

Sect. 4.2. We also compare our results with earlier research

findings.

• According to the results of LR analysis, the NDC has

significant effect on understandability prediction of

conceptual schema. The machine learning methods

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Area under ROC curve(AUC)=0.71

Sensitivity vs. 1-specificity

Sen

sitiv

ity

1-specificity

Fig. 4 ROC curve for

performance of LR analysis

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0



Sen

sitiv

ity

1-specificity


performance of Naive Bayesian

Classifier


123

(Naive BC, DT) also confirmed the findings of regres-

sion analysis as the values of sensitivity, specificity and

accuracy for NDC metric are high and same as of LR

analysis. Therefore, we reject the null hypothesis and

accept the alternative hypothesis (H1). However, Ser-

rano et al. (2007) found out that NDC metric has not a

significant relationship with understandability of con-

ceptual schema.

• According to the results of LR analysis, the NBC

metric has not a significant effect on understandability

prediction of conceptual schema. The machine learning

methods (Naive BC, DT) also confirmed the findings of

regression analysis as the values of sensitivity is low

than that of LR analysis. Hence, we reject the alternate

hypothesis H2 and accept the null hypothesis. However,

Serrano et al. (2007) found out that NBC metric has


ceptual schema.

• According to the results of LR analysis, the NC has


conceptual schema. The Naive BC method also

confirmed the findings of regression analysis as value

of sensitivity is high. The DT method also offers high

specificity as of LR analysis. Both Naive BC and DT

methods offer higher accuracy than that of LR analysis.

Serrano et al. (2007) also confirms that NC metric has


ceptual schema. Therefore, we reject the null hypoth-

esis and accept the alternate hypothesis H3.

• According to the results of LR analysis, the RBC metric

has not a significant effect on understandability

prediction of conceptual schema. The Naive BC

method also confirmed the findings of regression

analysis as the values of sensitivity, specificity and

accuracy are low. The DT method also offers the same

specificity as of LR analysis. Hence, we reject the

alternate hypothesis H4 and accept the null hypothesis.

However, Serrano et al. (2007) found out that NBC

metric has significant relationship with understandabil-

ity of conceptual schema.

• According to the results of LR analysis, the NAFC


prediction of conceptual schema. The DT method also

confirmed the findings of regression analysis as the

value of specificity is low. Hence, we reject the


Serrano et al. (2007) also found out that NAFC metric

has no significant relationship with understandability of

conceptual schema.

• According to the results of LR analysis, the NADC



methods (Naive BC and DT) also offers less or equal

specificity as of LR analysis. Serrano et al. (2007) also

found out that NADC metric has no significant

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0



Sen

sitiv

ity

1-specificity


performance of decision tree

method

Table 10 Prediction methods comparison

Method Sensitivity Specificity Accuracy AUC

LR analysis 0.6363 0.5555 0.60 0.71

Naı̈ve Bayesian

Classifier (BC)

0.7272 0.7777 0.75 0.889

J48 Decision

Trees (DT)

0.8181 0.6666 0.75 0.773


123

relationship with understandability of conceptual

schema. Hence, we reject the alternate hypothesis H6

and accept the null hypothesis.

• According to the results of LR analysis, the NABC



methods (Naive BC, DT) also confirmed the findings of

regression analysis as values of sensitivity and accuracy

are low. Serrano et al. (2007) also found out that NABC

metric has no significant relationship with understand-

ability of conceptual schema. Hence, we reject the


• According to the results of LR analysis, the NA has


conceptual schema. The machine learning methods

(Naive BC, DT) also confirmed the findings of regression

analysis as values of sensitivity, specificity and accuracy

are high. Serrano et al. (2007) also confirms that NA

metric has significant relationship with understandability

of conceptual schema. Therefore, we reject the null

hypothesis and accept the alternate hypothesis H8.

• According to the results of LR analysis, the NH metric




values of specificity and accuracy are low. Therefore,

we reject the alternate hypothesis H9 and accept the null

hypothesis. However, Serrano et al. (2007) found that

NH metric has significant relationship with understand-

ability of conceptual schema.

• According to the results of LR analysis, the DHP metric




value of specificity is low. Therefore, we reject the


However, Serrano et al. (2007) found that NH metric

has significant relationship with understandability of

conceptual schema.

• According to the results of LR analysis, the RSA metric




value of sensitivity is low. Serrano et al. (2007) also

found that RSA metric has no significant relationship

with understandability of conceptual schema. There-

fore, we reject the alternate hypothesis H11 and accept

the null hypothesis.

• The NDC and NC metrics are included in the multi-

variate analysis method, i.e. NDC and NC metrics in

combination are best predictors for understandability of

conceptual schema. Therefore, we accept the alternate

hypothesis H12 and reject the null hypothesis.

In some cases, we found some contradictions with ear-

lier research findings of other authors. These contradictions

may happen because of the various threats to the validity of

results (for example, threats to construct, internal, external

and conclusion validity), as discussed in the next secton.

9 Threats to validity of results

In this section, we discuss threats to construct, internal,

external and conclusion validity.

• Construct validity The construct validity is the degree

to which the independent and the dependent variables

are accurately measured by the measurement instru-

ments used in the study. We use the dependent variable

as understanding time i.e., the time each subject taken

to perform given tasks, so we consider that variable is

constructively valid. The construct validity of the

measures used for the independent variables (Structural

Metrics) is guaranteed by the Distance framework

(Poels and Dedene 1999) used for their theoretical

validation.

• Internal validity The internal validity is the degree to

which conclusions can be drawn about the causal effect

of independent variables on the dependent variables.

The following issues should be considered:

• Differences among subjects Experiments with in the

subjects reduce variability among subjects.

• Differences among schemas The considered sche-

mas had different domains. This could affect the

results obtained in some way.

• Precision of time recorded The subjects were asked

to record the start and finish times of each given

test. In our view, this method is more effective than

deploying another person to records the time of

each subject. However, we also understand that the

subject could introduce some imprecision while

recording the time.

• Learning effects The tests were ordered and given in

a distinct order for different subjects. So, each

subject answered the tests in the given order. In

doing this, we tried to minimize learning effects.

• Fatigue effects The average time for completing the

experiment was 16.85 min varying from a mini-

mum of approximately 12.30 min and a maximum

of about 22.21 min. With this range of times we

believe that fatigue effects hardly exist at all. Also,

the different order of the tests helped to avoid these

fatigue effects.

• Persistence effects In our case, persistence effects

were not present because the subjects had never

participated in a similar experiment.


123

• Motivation among subjects Subjects were volun-

teers and they were convinced that they were doing

useful exercises. The subjects wanted to participate

in the experiment. Therefore, we believe that

subjects were motivated at doing the experiment.

• Plagiarism and influence among subjects In order to

avoid these effects a supervisor was present during

the experiment. Subjects were informed they should

not talk to each other and they will not share their

answers with other subjects.

• External validity The external validity is the degree to

which the results of the research can be generalized to

the population under study and to other research

settings. The greater the external validity, the more

the results of an empirical study can be generalized to

actual software engineering practice. Two threats to

validity have been identified which limit the ability to

apply any such generalization:

• Materials and tasks used We tried to use schemas

and operations representative of real world cases in

the experiments, although more experiments with

larger and more complex schemas are necessary.

• Subjects Although this experiment was run by M.

Tech. students, we understand that the number of

subjects, in our case 18 subjects could be insuffi-

cient for generalizing the results obtained. More

experiments with practitioners and professionals

must be carried out in order to be able to generalize

the results.

• Conclusion validity The conclusion validity defines the

extent to which conclusions are statistically valid. The

only issue that could affect the statistical validity of this

study is the size of the sample data (20 values), which we

feel is not enough for both parametric and non-paramet-

ric statistic tests (Briand et al. 1995). We will try to obtain

bigger sample data through more experimentation.

10 Conclusions and future work

In this paper, we performed the experiment where 18 M.

Tech. (Engineering Postgraduate) students were given 20

conceptual schemas of DW. They analyzed them and

answer the given questions and recorded the time (under-

standing time) taken for each schema. We applied statis-

tical and machine learning methods on the collected data to

predict the effect of (individual or combined) schema

metrics on its understandability. The analysis results of all

these methods have been presented in Sects. 6 and 7. All

the proposed hypothesis have been successfully validated,

as already presented in the Sect. 8. Following are the points

worth notable based on the results of all the three predic-

tion methods (LR analysis, DT, Naive BC):

• The NDC, NC and NA metrics have significant effect

on understandability of conceptual schema of DW. It

has also been shown that the NDC metric has highest

sensitivity, highest specificity and highest accuracy

amongst all the metrics. Therefore, we accepted the

alternate hypothesis corresponding to these three met-

rics and rejected the null hypothesis.

• The remaining metrics (NBC, RBC, NAFC, NADC,

NABC, NH, DHP, RSA) are not to found to have

significant effect on understandability of conceptual

schema. Therefore, we rejected the alternate hypothesis

for these metrics and accepted the null hypothesis.

• LR analysis results showed that the NDC and NC have

combined effect on understandability of conceptual

schema.

• We have evaluated the performance of each prediction

method on the basis of receiver operating characteris-

tics (ROC) analysis. The performance comparison of

prediction methods (ref. Sect. 7) showed that Naive

Bayes Classifier method has predicted understandabil-

ity more accurately, as the value of the area under ROC

cure (AUC) is highest (0.889).

We only considered 20 conceptual schemas and

involved only 18 subjects (students), with this small data, it

seems difficult to generalize the results obtained in our

study. In near future, we will consider more number of

conceptual schemas and more number of subjects to rep-

licate the experiment. We would also apply more machine

learning methods such as SVM, Random forest etc. to

validate the effect of structural metrics on understand-

ability. This will help to generalize the results obtained.

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